Return to my Mathematics pages
Go to my home page
The Ellipse
© Copyright 2002, Jim Loy
The ellipse (left) is a
Conic Section. That means that we can
produce one by slicing a cone with a plane. It is kind of a flattened circle,
or a circle is a kind of regular ellipse. In fact, when you look at a circle
from some odd angle, you actually see an ellipse. And there is one way to look
at any ellipse so that it looks like a circle. On the right we see the famous
way to draw an ellipse, using string and a pencil. The ends of the string are
attached to fixed points called the foci (singular = focus) of the ellipse. The
point of the pencil is restrained by the string, and draws the ellipse. I have
shown the length of the string, below the ellipse. Usually we use a loop of
string, so we can smoothly draw the entire ellipse. The two red lines in the
same diagram are called the semi-major axis (a) and the semi-minor axis (b).
The eccentricity of the ellipse (a measure of its shape) is e =
sqrt(a-b^2/a^2), where sqrt() is the square root function, and b^2 is b
squared.
On the left is the traditional way to
define an ellipse, as a conic section. It is the set of points whose distance
from a point (the focus) and a line (the directrix) is in a constant proportion
(for the ellipse in the diagram, the proportion is 3/4). A proportion of 1
gives a parabola, and greater than 1 gives a hyperbola.
There are several ways to draw an ellipse. One equation (in analytic
geometry) is x^2/a^2 + y^2/b^2 = 1 (where x^2 means x squared). If a and b are
equal, the ellipse is a circle.
The area of an ellipse is A=pi ab. You may guess that there is a simple
formula for the circumference of an ellipse, but you would be wrong. The arc
length can be approximated in a number of ways, to any desired accuracy. For
example, integral calculus provides one fairly obvious way. The old way is to
use elliptical integrals, which cannot in general be solved exactly, and must
be approximated. In fact older mathematical table books contained tables of
elliptical integrals. A good approximation of the circumference is (where
sqrt() is the square root function):
pi sqrt(2(a^2+b^2))
If your browser supports Java, here are some animations involving
ellipses:
Also, see The Astroid.
Addendum:
I received email asking if we knew the length of the
major axis, the position of one end of the major axis, and the position of one
other point, would that define one and only one ellipse. Well, here are four
such ellipses, and it is obious that there are infinitely many such
ellipses.
It takes five points to determine a Conic Section (I don't know who proved that
originally). Most of these are ellipses and hyperbolas. A few specific conic
sections can be determined with fewer points (a circle requires three points).
But most require all five points, or some other conditions.
In Analytic Geometry, we often
rotate the ellipse (and maybe move it), or rotate the coordinate system (which
is the same thing), in order to simplify the equations.
Return to my Mathematics pages
Go to my home page